The displacers in most prior art Vuilleumier heat pumps are driven by a crank, such as shown in U.S. Pat. No. 1,275,507. A schematic of such a heat pump with crank driven displacers is shown in FIG. 1. In the '507 patent, the displacers have a phase difference of 90 degrees as shown in FIG. 2. A mechatronically-driven Vuilleumier heat pump, which is commonly assigned to the assignee of the present disclosure, has been disclosed in WO 2013/155258. In such a heat pump, the displacers are independently actuated allowing one displacer to remain stationary while the other displacer moves, which provides many additional degrees of freedom in controlling displacer motion. In the WO 2013/155258 A1 publication, a three-process cycle is also disclosed. A cycle that provides a high coefficient of performance is desired.